Theorem oev 7594

Description: Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
oev	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = if(𝐴 = ∅, (1𝑜 ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oev

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2		oveq2 6658	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·𝑜 𝑦) = (𝑥 ·𝑜 𝐴))
3	2	mpteq2dv 4745	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)))
4		rdgeq1 7507	. . . . 5 ⊢ ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜) = rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜))
5	3, 4	syl 17	. . . 4 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜) = rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜))
6	5	fveq1d 6193	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧))
7	1, 6	ifbieq2d 4111	. 2 ⊢ (𝑦 = 𝐴 → if(𝑦 = ∅, (1𝑜 ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧)) = if(𝐴 = ∅, (1𝑜 ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧)))
8		difeq2 3722	. . 3 ⊢ (𝑧 = 𝐵 → (1𝑜 ∖ 𝑧) = (1𝑜 ∖ 𝐵))
9		fveq2 6191	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
10	8, 9	ifeq12d 4106	. 2 ⊢ (𝑧 = 𝐵 → if(𝐴 = ∅, (1𝑜 ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧)) = if(𝐴 = ∅, (1𝑜 ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
11		df-oexp 7566	. 2 ⊢ ↑𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1𝑜 ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧)))
12		1on 7567	. . . . 5 ⊢ 1𝑜 ∈ On
13	12	elexi 3213	. . . 4 ⊢ 1𝑜 ∈ V
14		difss 3737	. . . 4 ⊢ (1𝑜 ∖ 𝐵) ⊆ 1𝑜
15	13, 14	ssexi 4803	. . 3 ⊢ (1𝑜 ∖ 𝐵) ∈ V
16		fvex 6201	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V
17	15, 16	ifex 4156	. 2 ⊢ if(𝐴 = ∅, (1𝑜 ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) ∈ V
18	7, 10, 11, 17	ovmpt2 6796	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = if(𝐴 = ∅, (1𝑜 ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  ifcif 4086   ↦ cmpt 4729  Oncon0 5723  ‘cfv 5888  (class class class)co 6650  reccrdg 7505  1_𝑜c1o 7553   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oexp 7566

This theorem is referenced by:  oevn0  7595  oe0m  7598